Computation of Lyapunov values for two planar polynomial differential systems

The complex formulas of computing the Lyapunov values are investigated for two planar polynomial systems with a pair of pure imaginary eigenvalues. The relations between the two systems and their standard forms are discussed and the transformation formulas are given. The corresponding complex formulas of computing the Lyapunov values are obtained. The results obtained here can be applied to judging the type of singular points and the number of limit cycles in planar polynomial systems.